Transformer resonant converter

ABSTRACT

Some embodiments may include a nanosecond pulser comprising a plurality of solid state switches; a transformer having a stray inductance, L s , a stray capacitance, C s , and a turn ratio n; and a resistor with a resistance, R, in series between the transformer and the switches. In some embodiments, the resonant circuit produces a Q factor according to 
     
       
         
           
             
               Q 
               = 
               
                 
                   1 
                   R 
                 
                  
                 
                   
                     
                       L 
                       s 
                     
                     
                       C 
                       s 
                     
                   
                 
               
             
             ; 
           
         
       
     
     and the nanosecond pulser produces an output voltage V out  from an input voltage V in , according to V out =QnV in .

BACKGROUND

Producing high voltage pulses with fast rise times is challenging. Forinstance, to achieve a fast rise time (e.g., less than about 50 ns) fora high voltage pulse (e.g., greater than about 10 kV), the slope of thepulse rise must be incredibly steep. Such a steep rise time is verydifficult to produce. This is especially difficult using standardelectrical components in a compact manner. It is additionally difficultto produce such a high voltage pulse with fast rise times havingvariable pulse widths and/or a variable high pulse repetition rate.

SUMMARY

Systems and methods are disclosed for producing high voltage, highfrequency pulses using a switching voltage source and a transformer thatincludes a resonant converter such as, for example, a series resonantconverter.

Some embodiments may include a resonant converter comprising a DC input,a plurality of solid state switches (which for us might be comprised ofthe SPA, a switching power amplifier based on the full bridge topology);a transformer having a stray inductance, L_(s), a stray capacitance,C_(s), and a primary to secondary turns ratio n; a total seriesresistance, R, that will be comprised of the stray series circuitresistance, R_(s), and any additional series resistance, R_(a), that isintentionally added to control Q; a diode rectifier on the secondaryside of the transformer; and an output waveform filter. In someembodiments, the resonant circuit has a Q factor according to

${Q = {\frac{1}{R}\sqrt{\frac{L_{s}}{C_{s}}}}};$

and the resonant converter produces an output voltage V_(out) from aninput voltage V_(in), according to V_(out)=QnV_(in). In someembodiments, the stray inductance is measured from the primary side ofthe transformer and the stray capacitance is measured from the secondaryside. In some embodiments, additional capacitance, C_(a), and/orinductance, L_(a), may be included to produce a desired resonantfrequency and/or change the circuit Q.

Some embodiments may include a resonant converter circuit having atransformer having a stray inductance, L_(s), and a stray capacitance,C_(s); and a stray resistance with a resistance, R_(s), in series withthe transformer. In some embodiments, the resonant circuit produces a Qfactor according to

${Q = {\frac{1}{R}\sqrt{\frac{L}{C}}}},$

where R is the sum of the series stray resistance R_(s) and anyadditionally added resistance R_(a) and equivalent series loadresistance R_(L), C is the sum of the stray capacitance C_(s) and anyadded capacitance C_(a) and any other stray capacitance C_(so), and L isthe sum of the stray series inductance L_(s) and any additional addedinductance L_(a) and any other stray series inductance L_(so).

In some embodiments, the output can have a voltage greater than 5 kV, 15kV, and/or 50 kV.

In some embodiments, the resonant converter can operate with a frequencygreater than about 25 kHz or 100 kHz.

In some embodiments, the ratio between a peak output power and anaverage output power is greater than a factor of 10.

In some embodiments, the stray inductance L_(s) comprises more than 50%of the total circuit inductance.

In some embodiments, the output pulses have a rise time with a voltageslew rate greater than 10⁹ V/s.

In some embodiments, the resonant converter includes an output that isgalvanically isolated from its input (e.g., a floating output).

In some embodiments, the pulse output voltage can be adjusted during thepulse duration with a timescale of less than 10 μs to make theadjustment to a new voltage output level.

In some embodiments, the stray capacitance C_(s) comprises more than 50%of the total circuit resonant capacitance.

In some embodiments, the peak output power is greater than 5 kW orgreater than 50 kW.

The embodiments described in this document, whether in the summary orelsewhere, are mentioned not to limit or define the disclosure, but toprovide examples to aid understanding thereof. Additional embodimentsare discussed in the Detailed Description, and further description isprovided there. Advantages offered by one or more of the variousembodiments may be further understood by examining this specification orby practicing one or more embodiments presented.

BRIEF DESCRIPTION OF THE FIGURES

FIG. 1 is an example transformer resonant converter according to someembodiments.

FIG. 2 is a circuit diagram of an example transformer resonant convertercoupled with switching circuitry and a load according to someembodiments.

FIG. 3 is a photograph of an example resonant converter.

FIG. 4 is an example waveform created from a transformer resonantconverter according to some embodiments.

FIG. 5 is an example waveform created from a transformer resonantconverter according to some embodiments.

FIG. 6 is an idealized example of a series resonant circuit according tosome embodiments.

FIG. 7 is a circuit diagram of an example transformer resonant converteraccording to some embodiments.

DETAILED DESCRIPTION

Systems and methods are disclosed for producing high voltage, highfrequency pulses using a switching voltage source and a transformer,arranged with other components, to be a series resonant converter, or atransformer resonant converter. The switching voltage source, forexample, may include a full bridge or a half bridge topology. Forexample, the switching voltage source may include a full bridge or ahalf bridge switch topology. As another example, the switching voltagesource may have additional output filter elements. The switching voltagesource, for example, may include a full bridge topology or a half bridgetopology. In some embodiments, the switching voltage source may includea switching power amplifier.

The transformer resonant converter, for example, may not include anyphysical capacitors and/or inductors. Instead, in some embodiments, thetransformer resonant converter may include a resistor in series with thestray capacitance and/or the stray inductance of at least thetransformer. The stray inductance, L_(so), and/or stray capacitance,C_(so), of other circuit elements may also be leveraged as part of theresonant converter. In some embodiments, the total stray inductanceand/or the stray capacitance can be small. For example, the strayinductance can be less than about 3,000 nH, 300 nH, 30 nH, 3 nH etc., asmeasured on the primary side of the transformer. As another example, thestray capacitance can be less than about 300 pF or less than about 30pF, as measured on the secondary side of the transformer. Additionalcapacitance, C_(a), and inductance, L_(a), may be added in conjunctionsuch as, for example, in parallel and/or series with the straycapacitance and stray inductance.

Resonant converters typically leverage the resonance of a circuit whenthe circuit is driven at the resonant frequency (an example seriesresonant circuit is shown in FIG. 6). The resonant frequency can bedetermined from the total inductance and capacitance of the circuitelements, for example, from the following:

$\begin{matrix}{f = \frac{1}{2\; \pi \sqrt{LC}}} & (1)\end{matrix}$

in this example, L and C represent the total effective and/or equivalentseries circuit inductance and capacitance, respectively, and as definedabove, L=L_(s)+L_(so)+L_(a), and C=Cs+C_(so)+C_(a). FIG. 6 shows anidealized series resonant circuit 600 without any resistance; and withan inductor 620, a capacitor 610, and power source 605. Resistance maybe present in various forms throughout the circuit.

When a resonant circuit is driven at its resonant frequency theeffective reactance of each of the circuit components are equal inmagnitude but opposite in sign. Therefore, they cancel each other outand all that's left is the real resistance of the circuit whethercomposed of stray resistance and/or resistive elements, including theload. In some cases, this real resistance can be the resistance of thecopper traces and/or any other circuit components in series with theresonant LC components. The ratio of the reactive components to the realresistance is defined as the Q factor, which is a dimensionlessparameter that is a good estimate for what multiplier the drivingvoltage will ring up to when measured across either L or C. The resonantfrequency and Q factor can be calculated from the following:

$\begin{matrix}{Q = {\frac{1}{R}{\sqrt{\frac{L}{C}}.}}} & (2)\end{matrix}$

R is the total equivalent series/dissipative resistance and may includeany series stray resistance Rs as well as any additionally addedresistance Ra, and additional equivalent series load resistance R_(L),as well as any other dielectric or other dissipative losses fromswitches or other components. Typical resonant converters use discretephysical circuit components for the inductor, capacitor, and/or aresistor to create a desired Q factor and resonant frequency f. In someembodiments, additional resistance may be left out to improve circuitefficiency. In some embodiments, some form of feedback and control maybe used to regulate the output voltage to a value lower than that whichwould naturally be set by the circuit Q. One such form of feedback andcontrol, for example, may rely on pulse width modulation of theswitching voltage source.

FIG. 1 is a circuit diagram of an example transformer resonant converter100 according to some embodiments. On the primary side 160 of thetransformer, the resonant converter 100 may include, for example, a DCinput 105 coupled with a switch 110. In some embodiments, the switch mayinclude a freewheeling diode or a body diode. The switch 110 may openand close at high frequencies, such as, for example, at the resonantfrequency of the transformer resonant converter 100.

The switch 110, for example, may be any type of solid-state switch. Theprimary side 160 of the resonant converter may also include resonantseries inductance 115 and resonant series resistance 120. The switch110, for example, can produce high frequency pulses such as atfrequencies greater than 50 KHz, 500 kHz, 5000 KHz, for example.

In some embodiments, the switch 110 may operate with transition timesless than, for example, about 40 ns, 10 ns, or 1 ns.

In some embodiments, the switch 110 may include a solid-state switch.The switch 100, for example, may include an IGBT switch, MOSFET switch,FET switch, GaN switch, etc. In some embodiments, the switch 110 may bea high efficiency switch. In some embodiments, the switch 110 may be afast switch (e.g., switching with a frequency greater than 100 kHz),which may allow for an output with low ripple. In some embodiments,pulse width modulation (PWM) techniques can be utilized for fast controlof the output voltage, to allow, for example, the control of beamcharacteristics with tens of μs resolution, for example, when drivingneutral beams.

The resonant series inductance 115 may include, for example, strayinductance of the transformer, stray inductance of the primary side 160circuitry, and/or a physical inductor. The resonant series inductance115 may be small, for example, less than about 3,000 nH, 300 nH, 30 nH,3 nH, etc.

The resonant series resistance 120 may include stray resistance and/or aphysical resistor. In some embodiments, a physical resistor may lowercircuit efficiency, however, a physical resistor may allow for fastercircuit response times, and/or may reduce the need for feedback andcontrol loops to control/regulate the output voltage.

The transformer 125 may include any type of transformer such as, forexample, a toroid shaped transformer with one or more primary sidewindings and a plurality of secondary side windings. As another example,the transformer 125 may be a coaxial transformer with one or moreprimary side windings and a plurality of secondary side windings. Insome embodiments, the one or more primary side windings may include aconductive sheet. In some embodiments, the one or more secondarywindings may include a conductive sheet.

The circuitry on the secondary side of the transformer 125 may includeresonant series capacitance 130. The resonant series capacitance 125,for example, may include stray capacitance of the transformer and/orstray capacitance of the secondary side circuitry and/or a capacitor.The resonant series capacitance 125 may be small, for example, less thanabout 1,000 pF, 100 pF, 10 pF, etc. The resonant series capacitance 125may be in parallel with the transformer output. The secondary side 165of the circuit may include a rectifier 135 and/or an output filter 140.

In some embodiments, a primary winding and/or a secondary winding mayinclude single conductive sheet that is wrapped around at least aportion of a transformer core. A conductive sheet may wrap around theoutside, top, and inside surfaces of a transformer core. Conductivetraces and/or planes on and/or within the circuit board may complete theprimary turn, and/or connect the primary turn to other circuit elements.In some embodiments, the conductive sheet may comprise a metal sheet. Insome embodiments, the conductive sheet may comprise sections of pipe,tube, and/or other thin walled metal objects that have a certaingeometry.

In some embodiments, a conductive sheet may terminate on one or morepads on a circuit board. In some embodiments, a conductive sheet mayterminate with two or more wires.

In some embodiments, a primary winding may include a conductive paintthat has been painted on one or more outside surfaces of the transformercore. In some embodiments, the conductive sheet may include a metalliclayer that has been deposited on the transformer core using a depositiontechnique such as thermal spray coating, vapor deposition, chemicalvapor deposition, ion beam deposition, plasma and thermal spraydeposition, etc. In some embodiments, the conductive sheet may comprisea conductive tape material that is wrapped around the transformer core.In some embodiments, the conductive sheet may comprise a conductor thathas been electroplated on the transformer core. In some embodiments, aplurality of wires in parallel can be used in place of the conductivesheet.

In some embodiments, an insulator may be disposed or deposited betweentransformer core and the conductive sheet. The insulator, for example,may include a polymer, a polyimide, epoxy, etc.

The rectifier 135 may include any type of rectifier such as, forexample, a diode-based rectifier, a full-bridge rectifier (e.g., asshown in FIG. 7), a half-bridge rectifier, a three-phase rectifier, avoltage multiplying rectifier, etc. Any other type of rectifier can beused.

The output filter 140 may include any type of filter. For example, theoutput filter 140 may include a high pass filter, a low pass filter, aband pass filter, etc.

Some embodiments may include a transformer resonant converter 100 thathas low stray inductance measured from the primary side 160. Low strayinductance may include inductance less than, for example, about 3,000nH, 300 nH, 30 nH, 3 nH, etc.

Some embodiments may include a transformer resonant converter 100 thathas low stray capacitance as measured from the primary side 160. Lowstray capacitance may include capacitance less than, for example, about1,000 pF, 100 pF, 10 pF, etc.

Some embodiments may include the transformer resonant converter 100 thatcan produce high average output power such as greater than, for example,about 3 kW, 100 kW, 3 MW. For short bursts the peak power output, forexample, may exceed 30 kW, 300 kW, 3 MW. Some embodiments may include atransformer resonant converter 100 that produces pulses with highvoltage such as greater than, for example, 5 kV, 25 kV, 250 kV, 2500 kV.Some embodiments may include a transformer resonant converter 100 thatproduces a high power burst operation with a peak power greater than 5times the average operating power of the converter. In some embodiments,the peak power output may be in excess of the average output power by afactor, for example, of 5, 50, 500.

In some embodiments, the transformer resonant converter 100 can producehigh voltage pulses with a fast rise time, for example, less than, forexample, about 10 μs, 1 μs, 100 ns, 10 ns, etc. for voltages greaterthan for example 5 kV, 30 kV, 100 kV, 500 kV, etc.

In some embodiments, the transformer resonant converter 100 can producean output pulse with low voltage ripple such as, for example, less thanabout 5%. Typical output voltage ripple may be less than, for example,15% or 0.5%.

In some embodiments, the transformer resonant converter 100 can operatewith pulse width modulation that may allow for greater control of theoutput waveform and/or allow for high efficiency power output. In someembodiments, the transformer resonant converter may include real timefeedback and control of the high voltage and/or power output. In someembodiments, the low stray inductance and/or low stray capacitance,and/or high frequency of operation can allow for this feedback loop tobe fast.

In some embodiments, the transformer resonant converter 100 maysignificantly increase the overall power density of a system. Forexample, the transformer resonant converter 100 could be used with anelectron tube driver for high-power radar systems and/or RF systems. Insome embodiments, the transformer resonant converter 100 may increasethe overall power density of the high-power radar systems and/or the RFsystems. Power densities may exceed, for example, 0.5 W/cm³, 5 W/cm³, 50W/cm³, or 500 W/cm³.

In some embodiments, the transformer resonant converter 100 may includeswitching components that are at low voltage in a standard H-bridgepower supply configuration with a hard ground reference. This may, forexample, remove the requirement of floating each module to high voltageas seen in the pulse step modulator.

In some embodiments, a transformer resonant converter may include highvoltage components that include a high voltage transformer andrectification diodes and other high voltage components. These componentscan, for example, be packaged for safe high voltage using oil, pottingor other methods. In some embodiments, some components may be in airwith appropriate stand-off to eliminate corona generation, arcing,and/or tracking.

In some embodiments of the transformer resonant converter, the output istransformer isolated, so the same system can provide either a floatingor ground referenced output and/or can be configured to provide either apositive or negative polarity. This may allow, for example, the samedesign to be utilized for any of the various high voltage grids of aparticular neutral beam injection design including, for example, eitherpositive or negative ion extraction and acceleration as well as ion andelectron suppressor grids.

In some embodiments, a resonant converter may produce the same powerlevels with a dramatic decrease in overall system size and/or controlcomplexity as compared to the pulse step modulators used currently forsmaller neutral beam injector systems.

In some embodiments, the resonant converter may be safe to arc-faultsdue the inherent series resonant behavior of the supply. The seriesresonant behavior of a resonant converter may have a supply impedancethat is matched to the load. When an arc occurs, for example, thismismatch can reduce the power flowing in the primary side 160 of thecircuit and the voltage on the secondary may fall, whereby the currentin the arc cannot continue to increase to the point of damage to thegrids.

In some embodiments, the transformer resonant converter may have verylittle energy stored in its output filter components. For example, thisstored energy may be less than, for example, about 10 J, 1.0 J, or 0.1J. The high frequency of operation allows this stored energy to beminimized. In some embodiments, minimizing this stored energy can beimportant. This energy, for example, can damage load components whenarcs occur.

In some embodiments, a transformer resonant converter may be modular. Insome embodiments, a transformer resonant converter may be easily scaledto higher output power levels making it a possible choice for largeneutral beam injector systems such as, for example, like those used atNSTX, DIII-D, or ITER. For example, power supplies with a transformerresonant converter can be added together with output arranged in seriesto easily increase the output voltage. Similarly, output current can beincreased by adding units in parallel on the primary as long as the highvoltage side is scaled to account for the increase current level.

FIG. 2 is a circuit diagram of a transformer resonant converter 200coupled with a load 250 according to some embodiments. The transformer225, for example, can have any number of turns. For example, thetransformer can have a turn ratio of n=1, n=30, n=50, n=100, etc. Thetotal series inductance is represented by an inductor circuit element205 (e.g., having an inductance less than about 3,000 nH, 300 nH, 30 nH,3 nH) on the primary side 260, which may be primarily composed of thestray inductance L_(s) of the transformer. The total series capacitanceis represented by a capacitor circuit element 210 (e.g., having acapacitance less than about 1,000 pF, 100 pF, 10 pF, etc.) on thesecondary side 265, which may be primarily composed of the straycapacitance C_(s) of the transformer. The stray inductance 205 and/orthe stray capacitance 210 can be of any value depending on the size,type, material, etc. of the transformer and/or the number of turns ofthe transformer. In this circuit, a primary resistor 215 may be includedin the circuit in series with the inductor 205 and/or the capacitor 210.In some embodiments, the primary resistor 215 may have a small value,such as, for example, less than 3,000 mOhms, 300 mOhms, 30 mOhms, 3mOhms.

In this example, the transformer resonant converter 200 includesswitching circuitry with four switch circuits 230. However, any numberof switch circuits can be used. Each switch circuit 230 may include asolid-state switch 235 with any number of circuit elements. Thesolid-state switch may include, for example, an IGBT switch, MOSFETswitch, FET switch, GaN switch, etc. Each switch circuit 230 may alsoinclude stray inductance represented by circuit element 240 and/or strayresistance represented by circuit element 245. Each switch circuit 230may also include a diode 255.

In this example, the secondary side of the transfer may also include afull bridge rectifier 260, an output filter 270, a load element 250(e.g., in a specific example, comprising an 86 k Ohm resistor), and/or afilter resistor 280 (e.g., in a specific example, comprising a 10 k Ohmresistor) that acts in conjunction with an external user load capacitor285 (e.g., in a specific example, comprising a capacitor of 30 pF). Inthe circuit shown, for example, no feedback and control regulation maybe required.

Any number of circuit elements combined in any configuration may followthe rectifier. For example, these other elements may include capacitive,inductive, and/or resistive filter components, and/or the externalloads.

In some embodiments, a transformer resonant converter (e.g., transformerresonant converter 100, transformer resonant converter 200, transformerresonant converter 700, etc.) can produce pulses with variousproperties. For example, a transformer resonant converter can producepulses with a voltage greater than about 30 kV. For example, atransformer resonant converter can produce pulses with a voltage greaterthan about 5 kV, 25 kV, 250 kV, or 2,500 kV. For example, a transformerresonant converter can produce pulses with a rise time to or a fall timefrom voltages greater than about 25 kV of less than about 300 μs, 30 μs,3 μs. For example, a transformer resonant converter can produce pulseswith a variable pulse width. For example, a transformer resonantconverter can produce pulses with a variable frequency. For example, atransformer resonant converter can produce pulses with a variablevoltage. For example, a transformer resonant converter can producepulses for a dielectric barrier discharge and/or neutral beam injectiondevices. For example, a transformer resonant converter can producepulses that have a pulse width of any duration such as, for example,ranging from about 1 μs to DC.

For example, a transformer resonant converter can produce pulses with apulse repetition rate greater than about 1 kHz for continuous operationat average power levels in excess of several kilowatts. For example, atransformer resonant converter can produce pulses having a pulserepetition frequency greater than about 1 kHz, 30 kHz, or 1000 kHz. Forexample, a transformer resonant converter can produce pulses havingpower greater than about 3 kW, 100 kW, or 3 MW.

In some embodiments, a transformer resonant converter can be housed in arack-mountable enclosure (e.g., standard 6U enclosure that is hasapproximate dimensions of 10″×17″×28″). In some embodiments, atransformer resonant converter may have a high power density, forexample, a power density that can exceed 0.5 W/cm³, 5 W/cm³, 50 W/cm³ or500 W/cm³.

In some embodiments, a transformer resonant converter may include anytype of solid state switches such as, for example, an IGBT, an FET, aMOSFET, a SiC junction transistor, a GaN switch, etc.

FIG. 3 is a photograph of an example transformer resonant converterincluding a transformer with windings 310 and a plurality of resistors305. The value of the cumulative resistance of the plurality ofresistors 305 can be determined from equation 2 for a give Q factor. Thetransformer resonant converter also includes a plurality of solid stateswitches 315 that are coupled with heat sinks. The solid-state switchescan be arranged, for example, in the full bridge topology in thisinstance. The transformer resonant converter also includes a pluralityof full-bridge rectifying diodes 320. Numerous other circuit elementscan also be included.

FIG. 4 is an example waveform created from a transformer resonantconverter according to some embodiments. In this example, the outputvoltage is greater than 30 kV, has a rise time of about 4 μs and a flattop width of about 12 μs.

FIG. 5 is another example waveform created from a transformer resonantconverter according to some embodiments. This waveform was produced bythe switching resonant converter shown in FIG. 2. In this example, theinput voltage to the transformer resonant converter was 600 V and theoutput pulse is 30 kV. In this example, the rise time is about 5 μs andthe flat top width is about 20 μs. These waveforms could have additionalrises, flat tops, and falls, depending on the modulation of the resonantconverter. This waveform shows one typical output pulse; various otheroutput pulses are possible. In some embodiments, the high power density,power, frequency, rise time, and/or voltage of the output of atransformer resonant converter can be unique. These attributes, forexample, may be enabled by the use of a transformer (and/or circuit)with low stray capacitance and/or low stray inductance that allows foroperation at high frequency, and the use of solid state switches thatoperate at high power with very fast transition times.

FIG. 7 is a circuit diagram of an example transformer resonant converter700 according to some embodiments. In this example, a transformer 705 iscoupled with and/or is part of a resonant converter topology where thetransformer 705 has a step-up voltage of n, which represents the ratioof turns of the primary winding to the turns of the secondary winding ofthe transformer 705. In this transformer, the stray inductance L_(s) isrepresented by inductor 715, and/or the stray capacitance C_(s) isrepresented by capacitor 720 of the transformer. These stray elementsare leveraged as part of the resonant converter 700. The strayinductance L_(so) and stray capacitance c_(so) of other circuit elementscan also be leveraged be used in conjunction with stray inductance L_(s)715 and capacitance C_(s) 720 to achieve the desired resonant frequencyf, and Q. Resistor 710 represents the additional resistance R_(a) thatmay be included on the primary side of the transformer. Once the strayinductance L_(s) and the stray capacitance C_(s) of the transformer areknown, and the total inductance and capacitance are known, even if theyare only comprised of stray elements, the resistance, for example,R_(pri), can be selected to produce a given Q factor using, for example,equation (2). In this example, the voltage on the secondary of thetransformer can be calculated from the following:

V _(out) =QnV _(c)  (3).

Thus, the voltage on the secondary of the transformer can be stepped upby the transformer by a factor of n multiplied by the resonant converterby a factor of Q.

In some transformer resonant converters, the total stray inductance andthe total stray capacitance, of the transformer and/or other circuitelements are kept low, for example, to produce a resonant oscillatingvoltage at high frequency, and an output voltage with fast rise timesand/or fast fall rise times. For example, the circuit can switch at highfrequencies such as, for example, at frequencies greater than 50 kHz,500 kHz, 5 MHz, for example. The low total stray inductance and the lowtotal stray capacitance of the transformer and/or other circuit elementsmay also, for example, be kept low to produce fast rectified rise times,faster than 100 μs, 10 μs, 1 μs, for example.

In some embodiments, the stray capacitance can be measured from thesecondary side of the transformer. Alternatively, the stray capacitancecan be measured from the primary side of the transformer, which is equalto the capacitance on the secondary side of the transformer times thesquare of the turns ratio n.

In some embodiments, the stray inductance can be measured from theprimary side of the transformer. Alternatively, the stray inductance canbe measured from the secondary side of the transformer, which is equalto the inductance on the primary side of the transformer times thesquare of the turns ratio n.

In some embodiments, the total equivalent series capacitance can bemeasured from the secondary side of the transformer. Alternatively, thetotal equivalent series capacitance can be measured from the primaryside of the transformer, which is equal to the total equivalent seriescapacitance on the secondary side of the transformer times the square ofthe turns ratio n.

In some embodiments, the total equivalent series inductance can bemeasured from the primary side of the transformer. Alternatively, thetotal equivalent series inductance can be measured from the secondaryside of the transformer, which is equal to the total equivalent seriesinductance on the primary side of the transformer times the square ofthe turns ratio n.

The term “substantially” means within 5% to 15% of the value referredto, or within manufacturing tolerances.

Numerous specific details are set forth herein to provide a thoroughunderstanding of the claimed subject matter. However, those skilled inthe art will understand that the claimed subject matter may be practicedwithout these specific details. In other instances, methods,apparatuses, or systems that would be known by one of ordinary skillhave not been described in detail so as not to obscure claimed subjectmatter.

The use of “adapted to” or “configured to” herein is meant as open andinclusive language that does not foreclose devices adapted to orconfigured to perform additional tasks or steps. Additionally, the useof “based on” is meant to be open and inclusive, in that a process,step, calculation, or other action “based on” one or more recitedconditions or values may, in practice, be based on additional conditionsor values beyond those recited. Headings, lists, and numbering includedherein are for ease of explanation only and are not meant to belimiting.

While the present subject matter has been described in detail withrespect to specific embodiments thereof, it will be appreciated thatthose skilled in the art, upon attaining an understanding of theforegoing, may readily produce alterations to, variations of, andequivalents to such embodiments. Accordingly, it should be understoodthat the present disclosure has been presented for-purposes of examplerather than limitation, and does not preclude inclusion of suchmodifications, variations, and/or additions to the present subjectmatter as would be readily apparent to one of ordinary skill in the art.

That which is claimed:
 1. A resonant converter comprising: a DC inputproviding an input voltage V_(in); a plurality of solid state switcheselectrically coupled with the DC input; a transformer having a strayinductance, L_(s), a stray capacitance, C_(s), and a turn ratio, n, thetransformer electrically coupled with the plurality of solid stateswitches; a resistor with a resistance, R, disposed in series betweenthe transformer and the switches; and a plurality of rectifier diodes;wherein the resonant circuit produces a Q factor according to${Q = {\frac{1}{R}\sqrt{\frac{L_{s}}{C_{s}}}}};$ and wherein theresonant converter produces output pulses with an output voltage V_(out)from an input voltage V_(in), according to V_(out)=QnV_(in).
 2. Theresonant converter circuit according to claim 1, wherein the outputpulses have a voltage greater than about 10 kV.
 3. The resonantconverter circuit according to claim 1, wherein the output pulses have afrequency greater than 25 kHz.
 4. The resonant converter circuitaccording to claim 1, wherein the resonant frequency is greater than 100kHz.
 5. The resonant converter circuit according to claim 1, wherein theoutput pulses have an output power greater than 5 kW.
 6. The resonantconverter circuit according to claim 1, wherein the output pulses havean output power greater than 50 kW.
 7. The resonant converter circuitaccording to claim 1, wherein the resonant converter has a switchingtransition time less than 40 ns.
 8. The resonant converter circuitaccording to claim 1, wherein the resonant converter circuit has a totalcircuit inductance less than about 300 nH as measured on the primary ofthe transformer.
 9. The resonant converter circuit according to claim 1,wherein the resonant converter circuit operates with a total circuitcapacitance of less than about 100 pF as measured on the secondary sideof the transformer.
 10. The resonant converter circuit according toclaim 1, wherein the output pulses have a rise time, with a voltage slewrate greater than 10⁹ V/s.
 11. The resonant converter circuit accordingto claim 1, wherein the resonant converter has a power density greaterthan 1 W/cm³.
 12. The resonant converter circuit according to claim 1,wherein the stray capacitance C_(s) comprises more than 50% of the totalcircuit resonant capacitance C.
 13. A resonant converter circuitcomprising: a transformer having a stray inductance, L_(s), wherein thestray inductance, L_(s), is not from an inductor; a stray capacitance,C_(s), wherein the stray capacitance, C_(s), is not from a capacitor;and a resistor with a resistance, R, in series with the transformer;wherein the resonant circuit produces a Q factor according to$Q = {\frac{1}{R}{\sqrt{\frac{L_{s}}{C_{s}}}.}}$
 14. The resonantconverter circuit according to claim 13, wherein the transformer has aturn ratio n and the resonant converter produces an output voltage,V_(out), from an input voltage V_(in), according to V_(out)=QnV_(in).15. The resonant converter circuit according to claim 13, wherein theresonant converter circuit produces a pulse with a voltage greater thanabout 10 kV.
 16. The resonant converter circuit according to claim 13,wherein the resonant converter operates at a resonant frequency greaterthan 0.1 MHz.
 17. The resonant converter circuit according to claim 13,wherein the resonant converter circuit produces pulses with a switchingtransition time less than 40 ns.
 18. The resonant converter circuitaccording to claim 13, wherein the resonant converter circuit has atotal circuit inductance less than about 300 nH as measured on theprimary of the transformer.
 19. The resonant converter circuit accordingto claim 13, wherein the resonant converter circuit operates with atotal circuit capacitance less than about 100 pF as measured on thesecondary side of the transformer.
 20. The resonant converter circuitaccording to claim 13, wherein the output pulses have a rise time, witha voltage slew rate greater than 10⁹ V/s.
 21. The resonant convertercircuit according to claim 13, wherein the resonant converter circuitproduces output pulses with a power density greater than 1 W/cm³. 22.The resonant converter circuit according to claim 13, wherein theresonant converter does not include an inductor.
 23. The resonantconverter circuit according to claim 13, wherein the resonant converterdoes not include a capacitor.
 24. The resonant converter circuitaccording to claim 13, wherein the ratio between a peak output power andan average output power is greater than a factor of
 10. 25. The resonantconverter circuit according to claim 13, wherein the stray inductanceL_(s) comprises more than 50% of the total circuit inductance.
 26. Theresonant converter circuit according to claim 13, wherein the outputpulses have a rise time with a voltage slew rate greater than 10⁹ V/s.